Business Mathematics - I

Step-by-Step Solution:

Given two points:

Step 1: Calculate the slope ($m$)

Step 2: Write the equation using point-slope form

Rearranging into standard form:

Final Answer: The equation of the straight line is $4x + y - 13 = 0$ (or $y = -4x + 13$).

Step-by-Step Solution:

Given demand function:

Step 1: Find quantity ($Q$) when $P = 20$$20 = 40 - 0.2Q$$

Step 2: Differentiate with respect to $P$

From $P = 40 - 0.2Q$:

Taking the reciprocal:

Step 3: Compute point price elasticity of demand ($\epsilon_d$)

In absolute magnitude:

Final Answer: The price elasticity of demand is $|\epsilon_d| = 1.0$ (unitary elastic demand).

Step-by-Step Solution:

Let the given roots be:

Step 1: Calculate the sum of the roots ($S$)

Step 2: Calculate the product of the roots ($P$)

Step 3: Form the quadratic equation

A quadratic equation with sum of roots $S$ and product of roots $P$ is given by:

Substituting $S = 4\sqrt{5}$ and $P = -25$:

Final Answer: The quadratic equation is $x^2 - 4\sqrt{5}x - 25 = 0$.

Step-by-Step Solution:

Given equation:

Step 1: Simplify both sides

Left-hand side:

Right-hand side:

Thus:

Step 2: Solve for $x$ using logarithms

Take natural logarithms on both sides:

Evaluating the numerical values:

Now solve for $x$:

(Note: If the right side was a misprint for $16 = 2^4$, then $4x - 2 = 4 \implies x = 1.5$; if $16 \times 2 = 32 = 2^5$, $4x - 2 = 5 \implies x = 1.75$. As printed, the exact mathematical solution is $x = \frac{2 + \log_2(21)}{4}$).

Final Answer: $x = \frac{2 + \log_2(21)}{4} \approx 1.598$.

Step-by-Step Solution:

Given system of linear equations:

Step 1: Eliminate $x$ and $y$ using equations (1) and (2)

Multiply equation (1) by 2:

Now subtract equation (2) from equation (4):

Step 2: Substitute $z = 3$ into the system

From equation (2):

From equation (3):

Step 3: Solve for $y$ and $x$

Substitute equation (5) into equation (6):

Now find $x$:

Step 4: Verification

• Eq (1): $2 + 2(1) + 3(3) = 2 + 2 + 9 = 13$ (Matches)
• Eq (2): $2(2) + 4(1) + 3 = 4 + 4 + 3 = 11$ (Matches)
• Eq (3): $3(2) + 2(1) + 2(3) = 6 + 2 + 6 = 14$ (Matches)

Final Answer: The solution is $x = 2$, $y = 1$, $z = 3$.

Step-by-Step Solution:

This question contains two comprehensive analytical components:

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Part 1: Least Squares Method for Company Yearly Income

Given Data:

Number of years ($N$) = $7$ (odd). Let the middle year $2020$ be taken as the origin, defining transformed variable $u = \text{Year} - 2020$.

Step 1: Determine Trend Line Coefficients

Since $\sum u = 0$:

The trend equation is:

Step 2: Estimate Income for 2024 and 2026

1. For Year 2024: $u = 2024 - 2020 = 4$$y_{2024} = 59 + \left(\frac{13}{7}\right)(4) = 59 + \frac{52}{7} = 59 + 7.4286 = \mathbf{66.43\text{ million Rs}}$$

2. For Year 2026: $u = 2026 - 2020 = 6$$y_{2026} = 59 + \left(\frac{13}{7}\right)(6) = 59 + \frac{78}{7} = 59 + 11.1429 = \mathbf{70.14\text{ million Rs}}$$

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Part 2: Open-Economy Keynesian Macroeconomic Model

Given:

• National Income Identity: $Y = C + I + G + X - M$
• Consumption function: $C = C_0 + 0.8Y_d = 90 + 0.8Y_d$
• Investment: $I = 500\text{m}$
• Government expenditure: $G = 700\text{m}$
• Exports: $X = 250\text{m}$
• Import function: $M = 0.3Y_d$
• Tax rate: $t = 0.4$, with $T = tY = 0.4Y$
• Disposable Income: $Y_d = Y - T = Y - 0.4Y = 0.6Y$

Step 1: Derive the Aggregate Expenditure Equation ($E$)

Substitute $Y_d = 0.6Y$ into $C$ and $M$:

Aggregate Expenditure:

Step 2: Determine Equilibrium National Income ($Y^*$)

Set $Y = E$:

Step 3: Determine Equilibrium Consumption ($C^*$) and Total Tax ($T^*$)

• Consumption ($C^*$):

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  $$C^* = 90 + 0.48(2,200) = 90 + 1,056 = \mathbf{1,146\text{ million Rs}}$$
• Total Tax ($T^*$):

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  $$T^* = 0.4(2,200) = \mathbf{880\text{ million Rs}}$$

Final Answer:

• Least Squares Trend Line: $y = 59 + 1.8571(\text{Year} - 2020)$
  • 2024 Income: Rs 66.43 million
  • 2026 Income: Rs 70.14 million
• Expenditure Equation: $E = 1540 + 0.30Y$
• Equilibrium Income ($Y^*$): Rs 2,200 million
• Equilibrium Consumption ($C^*$): Rs 1,146 million
• Total Tax ($T^*$): Rs 880 million

Step-by-Step Solution:

Given two points:

Step 1: Compute the slope ($m$)

Step 2: Use point-slope form

Substitute $(x_1, y_1) = (3, 6)$ and $m = \frac{1}{2}$:

Rearrange into standard form $Ax + By + C = 0$:

Final Answer: The equation of the straight line is $x - 2y + 9 = 0$ (or $y = \frac{1}{2}x + \frac{9}{2}$).

Step-by-Step Solution:

Given:

• Total investment / budget ($M$) = $\text{Rs } 90,000$
• Unit price of material $X$ ($P_X$) = $\text{Rs } 1,500$
• Unit price of material $Y$ ($P_Y$) = $\text{Rs } 1,800$

Step 1: Formulate the Budget Constraint Equation

Step 2: Determine the Slope

Expressing $Y$ as a function of $X$ ($Y$ on the vertical axis, $X$ on the horizontal axis):

The slope of the budget constraint line is:

Final Answer: The slope of the budget constraint is $-\frac{5}{6}$ (or approximately $-0.833$).

Step-by-Step Solution:

Given Data:

Number of years ($N$) = $7$ (odd). Let the origin be the middle year $2011$, defining transformed time variable $u = \text{Year} - 2011$.

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Step 1: Compute Trend Line Coefficients

The trend equation is $y = a + bu$. Since $\sum u = 0$:

The least squares trend line is:

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Step 2: Estimate Profit for the Year 2015

For Year 2015:

Final Answer:

• Equation of trend line: $y = 175.7143 + 5.3571(\text{Year} - 2011)$
• Estimated profit for 2015: Rs 197.14 million

Step-by-Step Solution:

Given the two points:

Step 1: Calculate the slope ($m$) of the line

Using the two-point slope formula:

Step 2: Use the point-slope form to find the equation

Substitute $(x_1, y_1) = (8, -3)$ and $m = -\frac{1}{3}$:

Multiply both sides by $3$:

Rearranging into the standard linear equation form $Ax + By + C = 0$:

Final Answer: The equation of the straight line is $x + 3y + 1 = 0$ (or in slope-intercept form, $y = -\frac{1}{3}x - \frac{1}{3}$).

Step-by-Step Solution:

Given function:

To find the value of $f(3)$, substitute $x = 3$ into the function:

Final Answer: $f(3) = 63$.

Step-by-Step Solution:

Given data:

• Independent variable ($x$) = Year of service
• Dependent variable ($y$) = Monthly income (in thousands Rs)
• Number of observations ($N$) = $7$

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Step 1: Compute the means $\bar{x}$ and $\bar{y}$$\bar{x} = \frac{\sum x}{N} = \frac{62}{7} \approx 8.8571$$

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Step 2: Compute regression slope ($b$) and intercept ($a$)

The least squares regression line is $y = a + bx$.

Now find intercept $a$:

Thus, the least squares trend line is:

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Step 3: Estimate monthly income for 13 years and 15 years

1. For $x = 13$ years:

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   $$y_{13} = 0.34426 + 1.09016(13) = 0.34426 + 14.17208 = 14.51634 \text{ (thousand Rs)}$$

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   $$\text{Estimated Income} \approx \text{Rs } 14,516.34$$

2. For $x = 15$ years:

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   $$y_{15} = 0.34426 + 1.09016(15) = 0.34426 + 16.3524 = 16.69666 \text{ (thousand Rs)}$$

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   $$\text{Estimated Income} \approx \text{Rs } 16,696.66$$

Final Answer:

• Equation of straight line: $y = 0.3443 + 1.0902 x$
• Estimated income for 13 years: Rs 14,516.34
• Estimated income for 15 years: Rs 16,696.66